Answer :
Let's analyze the given linear inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] step-by-step to determine which statements are true.
1. The slope of the line is -2:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In the given inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
- Thus, the statement saying the slope of the line is -2 is false.
2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line:
- For inequalities of the form [tex]\( y > mx + b \)[/tex] or [tex]\( y < mx + b \)[/tex], the boundary line is dashed because the inequality does not include the equality (i.e., it is a strict inequality).
- Therefore, this statement is true.
3. The area below the line is shaded:
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] indicates that the region above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is the solution region.
- So, the area below the line is not shaded.
- This statement is false.
4. One solution to the inequality is [tex]\( (0,0) \)[/tex]:
- Plugging [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 0 > \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ 0 > -2 \][/tex]
- Since this inequality holds true, [tex]\( (0,0) \)[/tex] is indeed a solution.
- Thus, this statement is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]:
- The y-intercept is the point where [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y = \frac{3}{4} x - 2 \)[/tex]:
[tex]\[ y = \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ y = -2 \][/tex]
- Therefore, the y-intercept is at [tex]\( (0, -2) \)[/tex].
- This statement is true as well.
So, the three true statements are:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- One solution to the inequality is [tex]\( (0,0) \)[/tex].
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
1. The slope of the line is -2:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In the given inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
- Thus, the statement saying the slope of the line is -2 is false.
2. The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line:
- For inequalities of the form [tex]\( y > mx + b \)[/tex] or [tex]\( y < mx + b \)[/tex], the boundary line is dashed because the inequality does not include the equality (i.e., it is a strict inequality).
- Therefore, this statement is true.
3. The area below the line is shaded:
- The inequality [tex]\( y > \frac{3}{4} x - 2 \)[/tex] indicates that the region above the line [tex]\( y = \frac{3}{4} x - 2 \)[/tex] is the solution region.
- So, the area below the line is not shaded.
- This statement is false.
4. One solution to the inequality is [tex]\( (0,0) \)[/tex]:
- Plugging [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[ 0 > \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ 0 > -2 \][/tex]
- Since this inequality holds true, [tex]\( (0,0) \)[/tex] is indeed a solution.
- Thus, this statement is true.
5. The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex]:
- The y-intercept is the point where [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y = \frac{3}{4} x - 2 \)[/tex]:
[tex]\[ y = \frac{3}{4} \cdot 0 - 2 \][/tex]
[tex]\[ y = -2 \][/tex]
- Therefore, the y-intercept is at [tex]\( (0, -2) \)[/tex].
- This statement is true as well.
So, the three true statements are:
- The graph of [tex]\( y > \frac{3}{4} x - 2 \)[/tex] is a dashed line.
- One solution to the inequality is [tex]\( (0,0) \)[/tex].
- The graph intercepts the [tex]\( y \)[/tex]-axis at [tex]\( (0,-2) \)[/tex].
